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One of the most important problems in many research fields is the development of reliable mathematical models with good predictive ability to simulate experimental systems accurately. Moreover, in some of these fields, as marine systems, these models play a key role due to the changing environmental conditions and the complexity and high cost of the infrastructure needed to carry out experimental tests. In this paper, a semiphysical modelling technique based on least-squares support vector machines (LS-SVM) is proposed to determine a nonlinear mathematical model of a surface craft. The speed and steering equations of the nonlinear model of Blanke are determined analysing the rudder angle, surge and sway speeds, and yaw rate from real experimental data measured from a zig-zag manoeuvre made by a scale ship. The predictive ability of the model is tested with different manoeuvring experimental tests to show the good performance and prediction ability of the model computed.

The availability of tools and methods to compute mathematical models for simulation purposes is of key importance, making the system identification field one of the highlights among the research topics in engineering and one of the most important stages in the control research area. Moreover, aspects that go from the high cost of practical implementations and tests in an open air environment to the complexity of the infrastructure needed to carry out experimental tests, and even the changing environmental conditions, call for the availability of these mathematical models with which new designs and ideas can be tested in simulation with high accuracy. In addition, the importance of the modelling stage is crucial since an inadequate model identification may yield large prediction errors. The literature on linear and nonlinear system identification is extensive and covers many areas of engineering research. For a short survey on some essential features in the identification area and a classification of methods the reader is referred to [

Among the number of techniques and methods on system identification, semiphysical modelling presents some interesting own characteristics. In this method, the prior knowledge about the application is used to develop a good model structure to be defined with raw measurements. The model structures defined are not physically complete, but allow for very suitable models to describe the behaviour of the systems involved, [

One of the most popular and widely used techniques in the artificial intelligence (A.I.) field for system identification is the artificial neural networks, such as multilayer perceptron (MLP); see, for example, [

Although there are not many results for system identification using SVM, we can find some interesting works such as the work in [

In this paper LS-SVM is used for the semiphysical modelling of a surface marine vessel. System identification of marine vehicles starts in the 70s with the works in [

For the above reason, in some practical situations, it is usual to employ simple vehicle models that, although they reproduce with less accuracy the dynamics of the vehicle, they show very good results and prediction ability for most of the standard operations; see, for example, [

We can find some works that employ neural networks to define the dynamics of a surface marine vehicle, such as [

Therefore, the key contributions of the present paper are twofold: (i) the mathematical nonlinear Blanke model of a scale ship is computed from experimental data collected from a 20/20 degree zig-zag manoeuvre with the LS-SVM technique; (ii) the prediction ability of the mathematical model is tested on an open air environment with different manoeuvres carried out with the scale ship. These tests allow checking the connection between the mathematical model and the ship, showing how this nonlinear model predicts with large accuracy the actual behaviour of the surface vessel. In this sense, the model can be used to design control strategies to predict the ship behaviour on a simulation environment before its implementation on the real vehicle. It is important to keep in mind, for the experimental results obtained in this paper, that the analytical properties of SVM can be compromised in stochastic problems because the noise generates additional support vectors. However, if the noise ratio is good and the amplitude is limited, the SVM can solve the problem as if it was deterministic [

The document is organized as follows. In Section

For the sake of completeness and clarity, in this section LS-SVM is briefly introduced. The notation and concepts of this section follow the explanation in [

In the above problem formulation

Equation (

In marine systems, the experimental tests can become costly in time and money, due to the need of deployment, calibration, and recovery of complex systems at sea. Therefore, the number of experimental tests that may be carried out are partially constrained by this reason among others, like environmental conditions, transportation of equipment, and so forth, to name but a few. In this sense, the availability of mathematical models, which describe the real systems accurately, is of utmost importance because most of these experimental tests may be carried out in simulation, predicting with high accuracy the real behaviour of the real systems and saving a number of practical tests.

There exists a wide range of different marine systems that require mathematical models. The problem arises when a detailed and trustworthy mathematical ship model is needed, since it requires the identification of a multitude of hydrodynamic parameters; see [

In many practical scenarios, it is very usual to employ simple models that predict the behaviour of real ships with large accuracy in most of the standard operations, like the Nomoto models [

Therefore, in the present work a nonlinear manoeuvring model based on second-order modulus functions is employed. The model used is the one proposed by Blanke [

The interest of this particular model resides in that, despite its relative simplicity, the most important nonlinear terms of the ship dynamics are taken into account. Furthermore, it is possible to compute a dynamic model for control purposes from the experimental data without the need of computing the hydrodynamic derivatives that define all the ship characteristics and its complete behaviour. Therefore, (

for

The estimates of the elements in vectors (

The structure of the mathematical model is known in advance, and elements in vectors (

The data used for the training of the LS-SVM algorithm were obtained by carrying out a 20/20 degree zig-zag manoeuvre, since it is a simple manoeuvre but enough to define the main characteristics of the ship dynamics. Once the model is defined with the above zig-zag data, its prediction ability must be compared with the real behaviour of the ship for the same commanded input data, namely, surge speed and rudder angle.

The vehicle used for the experimental tests is a scale model in a

Main parameters and dimensions of the real and the scale ships.

Parameter | Real ship | Scale ship |
---|---|---|

Length between perpendiculars ( |
74.400 m | 4.389 m |

Maximum beam ( |
14.200 m | 0.838 m |

Mean depth to the top deck ( |
9.050 m | 0.534 m |

Design draught ( |
6.300 m | 0.372 m |

Scale ship used in the experimental tests.

The 20/20 degree zig-zag manoeuvre to obtain the training data is carried out with a commanded surge speed of 2 m/s, during 90 seconds. The sampling time is 0.2 seconds, so 450 samples are measured. Figure

20/20 degree zig-zag manoeuvre. Yaw angle (solid line) and rudder angle (dashed line).

Now the LS-SVM algorithm for regression may be trained with these input and output data to compute the vectors defined in (

Note that the term

Surge speed measured in the zig-zag manoeuvre with the ship (solid line) and in simulation (dashed line).

Similarly, in Figure

Sway speed measured in the zig-zag manoeuvre with the ship (solid line) and in simulation (dashed line).

Finally, in Figure

Yaw rate obtained in the zig-zag manoeuvre with the ship (solid line) and in simulation (dashed line).

For comparison purposes, in Figure

Approximation errors in the surge speed (dashed line), sway speed (dotted line), and yaw rate (solid line).

The predictive ability of the model must be tested with different tests and manoeuvres. For this purpose two different manoeuvres are now undertaken. These tests are some turning manoeuvres (evolution circles) and a 10/10 degree zig-zag manoeuvre. The initial values of the effective surge speed, sway speed, and yaw rate used in the simulation tests are the same as those of the real ones to show clearly the connection between the real and the simulated systems.

The first validation test consists in two turning manoeuvres (evolution circles) for commanded rudder angles of ±20 deg. The test was run during 240 seconds for each of the turning manoeuvres. In Figures

Surge speed obtained in two turning manoeuvres with the ship (solid line) and in simulation (dashed line), (a) −20 deg and (b) +20 deg.

In Figures

Sway speed obtained in two turning manoeuvres with the ship (solid line) and in simulation (dashed line), (a) −20 deg and (b) +20 deg.

The yaw rate for the simulated and the actual systems can be studied in Figures

Yaw rate obtained in two turning manoeuvres with the ship (solid line) and in simulation (dashed line), (a) −20 deg and (b) +20 deg.

In Figures

Approximation errors in the surge speed (dashed line), sway speed (dotted line), and yaw rate (solid line) for the turning manoeuvres, (a) −20 deg and (b) +20 deg.

In this second test a 10/10 degree zig-zag manoeuvre is carried out to prove the prediction ability of the model. The manoeuvre is run during 90 seconds. In Figure

Surge speed obtained in a 10/10 degree zig-zag manoeuvre with the ship (solid line) and in simulation (dashed line).

In Figure

Sway speed obtained in a 10/10 degree zig-zag manoeuvre with the ship (solid line) and in simulation (dashed line).

Yaw rate obtained in a 10/10 degree zig-zag manoeuvre with the ship (solid line) and in simulation (dashed line).

Approximation errors in the surge speed (dashed line), sway speed (dotted line), and yaw rate (solid line).

In Figure

Therefore, it is clear that the nonlinear mathematical model defined for a surface marine vehicle with LS-SVM provides a satisfactory result which predicts with large accuracy the nonlinear dynamics of the experimental system and that it is suitable to be used for control purposes. Thus, this technique has the potential to be implemented for different kinds of marine vehicles in a simple and fast manner, avoiding many practical tests to define a reliable mathematical model and providing a very large prediction ability.

It would be interesting as future research to compare the results obtained in this work with the results that would be obtained using extreme learning machines (ELM) [

In this work, the nonlinear ship model of Blanke has been computed using experimental data obtained from a zig-zag manoeuvre test. A semiphysical modelling technique based on a least squares support vector machines algorithm has been applied to determine the parameters of the nonlinear model using the rudder angle, surge and sway speeds, and yaw rate as training data. It was shown that the model obtained fits the training data in a nice manner, showing the simulated system a behaviour very similar to that of the real ship. Furthermore, the prediction ability of the model was validated carrying out several experimental tests, like turning manoeuvres and zig-zags, demonstrating that the mathematical model can reproduce the actual ship dynamics with large accuracy in different manoeuvres. In addition, the model computed is suitable to be used for testing control algorithms in simulation, avoiding the execution of a large number of experimental tests.

Future work will aim at (i) extending the methodology developed to deal with models whose structures are not known in advance to capture all the features of the real ship, incorporating disturbances and environmental conditions, (ii) studying the performance of control algorithms for path following and tracking with the ship model defined in comparison with the results obtained for the real vehicle, and (iii) comparing the results obtained in this work with other different identification techniques, like the extreme learning machines (ELM).

The authors wish to thank the Spanish Ministry of Science and Innovation (MICINN) for support under Projects DPI2009-14552-C02-01 and DPI2009-14552-C02-02. The authors wish to thank also the National University Distance Education (UNED) for support under Project 2012V/PUNED/0003.